From: Henry Baker <hbaker1@pipeline.com> To: Michael Kleber <michael.kleber@gmail.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, March 14, 2012 10:52 AM Subject: Re: [math-fun] circular-arc splines, again
The proof a non-existence for n=3 needs to be tightened up a little. ----------------------------------------------------------------------------------------- A C1 configuration of circular arcs remains so under inversion about a point, other than possibly extending to infinity. Invert about one of the arc junctions. The two arcs meeting at that junction become parallel rays, one from a finite point P and extending infinitely far to the left, the other from a finite point Q and extending infinitely far to the right. The rays cannot lie on the same line, as then the two arcs would be part of the same circle. If a 3-arc configuration exists, then P and Q are joined by a single circular arc A. The directed tangents to A at P and Q are parallel, so A must undergo an angle change of 2πn. If n=0, A is a line segment, and this is impossible because the rays do not lie on the same line. If n is not zero, A is (one or more) complete circles, and P and Q coincide, again impossible for the same reason. -- Gene